Select The Correct Answer.Simplify The Following Expression: $\left(5^3\right)^5$A. $5^8$B. $5^{10}$C. $5^{-2}$D. $5^{15}$

===========================================================

Introduction

---

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression $\left(5^3\right)^5$ and explore the correct answer among the given options.

Understanding Exponential Notation

---

Before we dive into simplifying the expression, let's review the basics of exponential notation. When we write $a^b$, it means $a$ raised to the power of $b$. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. Similarly, $5^3$ means $5$ multiplied by itself $3$ times, which equals $125$.

Simplifying the Expression

---

Now that we have a solid understanding of exponential notation, let's simplify the expression $\left(5^3\right)^5$. To do this, we need to apply the power rule of exponents, which states that $\left(a^b\right)^c = a^{bc}$. In this case, we have $\left(5^3\right)^5$, so we can rewrite it as $5^{3 \times 5}$.

Applying the Power Rule

---

Using the power rule, we can simplify the expression further. We multiply the exponents $3$ and $5$ to get $15$. Therefore, the simplified expression is $5^{15}$.

Evaluating the Options

---

Now that we have simplified the expression, let's evaluate the options:

• A. $5^8$: This is not the correct answer, as we simplified the expression to $5^{15}$.
• B. $5^{10}$: This is also not the correct answer, as we simplified the expression to $5^{15}$.
• C. $5^{-2}$: This is not the correct answer, as we simplified the expression to $5^{15}$.
• D. $5^{15}$: This is the correct answer, as we simplified the expression to $5^{15}$.

Conclusion

---

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By applying the power rule of exponents, we can simplify complex expressions and arrive at the correct answer. In this article, we simplified the expression $\left(5^3\right)^5$ and evaluated the options to arrive at the correct answer, $5^{15}$.

Frequently Asked Questions

---

Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^b\right)^c = a^{bc}$.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to apply the power rule of exponents. This involves multiplying the exponents and simplifying the resulting expression.

Q: What is the correct answer for the expression $\left(5^3\right)^5$?

A: The correct answer is $5^{15}$.

Final Thoughts

---

Simplifying exponential expressions is a fundamental concept in mathematics, and it requires a solid understanding of exponential notation and the power rule of exponents. By applying the power rule, we can simplify complex expressions and arrive at the correct answer. In this article, we simplified the expression $\left(5^3\right)^5$ and evaluated the options to arrive at the correct answer, $5^{15}$.

=====================================================

Introduction

---

Exponential expressions are a fundamental concept in mathematics, and understanding them is crucial for students and professionals alike. In this article, we will provide a comprehensive Q&A guide on exponential expressions, covering various topics and concepts.

Q&A: Exponential Expressions

---

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that represents a quantity raised to a power. It is written in the form $a^b$, where $a$ is the base and $b$ is the exponent.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^b\right)^c = a^{bc}$. This means that when we raise a power to another power, we multiply the exponents.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to apply the power rule of exponents. This involves multiplying the exponents and simplifying the resulting expression.

Q: What is the difference between $a^b$ and $b^a$?

A: $a^b$ and $b^a$ are two different exponential expressions. $a^b$ means $a$ raised to the power of $b$, while $b^a$ means $b$ raised to the power of $a$. For example, $2^3$ means $2$ raised to the power of $3$, while $3^2$ means $3$ raised to the power of $2$.

Q: How do I evaluate an exponential expression with a negative exponent?

A: To evaluate an exponential expression with a negative exponent, you need to use the rule $a^{-b} = \frac{1}{a^b}$. This means that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent.

Q: What is the difference between $a^{-b}$ and $\frac{1}{a^b}$?

A: $a^{-b}$ and $\frac{1}{a^b}$ are two different ways of writing the same expression. $a^{-b}$ means $a$ raised to the power of $-b$, while $\frac{1}{a^b}$ means the reciprocal of $a$ raised to the power of $b$.

Q: How do I simplify an exponential expression with a zero exponent?

A: To simplify an exponential expression with a zero exponent, you need to use the rule $a^0 = 1$. This means that any base raised to the power of zero is equal to $1$.

Q: What is the difference between $a^0$ and $a^1$?

A: $a^0$ and $a^1$ are two different exponential expressions. $a^0$ means $a$ raised to the power of $0$, while $a^1$ means $a$ raised to the power of $1$. Since any base raised to the power of zero is equal to $1$, $a^0$ is equal to $1$, while $a^1$ is equal to $a$.

Q&A: Exponential Functions

---

Q: What is an exponential function?

A: An exponential function is a mathematical function that represents a quantity that grows or decays exponentially. It is written in the form $f(x) = a^x$, where $a$ is the base and $x$ is the input.

Q: What is the difference between an exponential function and a linear function?

A: An exponential function and a linear function are two different types of functions. A linear function is a function that represents a straight line, while an exponential function represents a curve that grows or decays exponentially.

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What is the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

Q&A: Exponential Equations

---

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves an exponential expression. It is written in the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the constant.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use logarithms. You can use the logarithmic function to rewrite the equation in a form that is easier to solve.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation and a linear equation are two different types of equations. A linear equation is an equation that involves a linear expression, while an exponential equation involves an exponential expression.

Conclusion

---

In conclusion, exponential expressions are a fundamental concept in mathematics, and understanding them is crucial for students and professionals alike. By reviewing the Q&A guide provided in this article, you should have a better understanding of exponential expressions, functions, and equations.

Final Thoughts

---

Exponential expressions are a powerful tool in mathematics, and they have many real-world applications. By mastering exponential expressions, you can solve a wide range of problems in fields such as science, engineering, and finance.